There is a bit of variation in these doors but usually this what you do.
On one side at the top will be a pelmet board with screws, mine come up through the bottom edge. Unscrew this and take the board out.
Close the door and up the top you will see the hanging/slidding mechanism... get a couple of spanners and loosen the bolts/nuts that hold your door on and then you should be able to take the door off.
You should not need to remove the slidding mechanism/track.
I am not sure this will help you as there is not much you can do to the cavity frame by taking out the door. Can you check that there are no gaps at the bottom of the wall from under the house? What about above in the ceiling. The mice may have built a home or hide in there but I am not convinced that they are entering your house through there.
Do you want a lone of my cat?????
Get the little critters
Chipman